Discrete-Time Quantum Walks

Updated 4 December 2025

Discrete-Time Quantum Walks are unitary quantum processes on graphs that use coin and shift operations to iteratively evolve the system.
They generalize classical random walks by leveraging quantum coherence and entanglement, resulting in phenomena like ballistic spreading and localization.
They are applied in simulating quantum transport, topological phenomena, and optimizing quantum algorithms, making them vital in quantum computation research.

A discrete-time quantum walk (DTQW) is a unitary quantum dynamical process on a graph, where the evolution is iterated in discrete time steps via sequences of coin and shift (or other structured) operators. DTQWs generalize classical random walks, but harness quantum coherence and entanglement, exhibiting ballistic spreading, localization, and rich spectral properties. They form the basis of a universal model of quantum computation, provide algorithmic speedups, and serve as simulators for quantum transport, topological phenomena, and field theory discretizations.

1. Formal Definitions and Primary Models

The canonical DTQW is defined on a Hilbert space $\mathcal{H} = \mathcal{H}_P \otimes \mathcal{H}_C$ , where $\mathcal{H}_P$ encodes the positions (vertices or sites) and $\mathcal{H}_C$ serves as the internal "coin" degree of freedom, typically spanned by a finite set representing possible steps. Each time step consists of:

A coin operation $C$ , typically a vertex-wise block-diagonal unitary matrix, mixing the internal degrees of freedom,
A conditional shift $S$ , which propagates amplitudes between sites dictated by the coin state.

A standard choice in 1D is

The one-step propagator is $U = S(C \otimes I_P)$ ; the dynamics is $|\psi_{t+1}\rangle = U|\psi_t\rangle$ (Godsil et al., 2017).

Several structurally distinct DTQW models arise:

(A) Arc-Reversal Model

State space: oriented arcs $\ket{u,v}$ on a finite undirected graph.
Coin: for each vertex $u$ , a unitary $C_u$ on outgoing arcs, compatible with a linear neighbor-order.
Shift (arc-reversal): $R\ket{u,v} = \ket{v,u}$ .
Evolution: $U_{ar}= R C$ .

(B) Shunt-Decomposition Model (for regular graphs)

State space: vertices $\ket{u}$ and coin register $\ket{j}$ indexing neighbor-shifts.
Shift: $\ket{u}\otimes \ket{j} \to \ket{P_j(u)}\otimes\ket{j}$ where each $P_j$ is a permutation extending neighbor orderings.
Coin: $C = I_n \otimes C_{coin}$ , $C_{coin}\in U(d)$ .
Evolution: $U_{sh}= S C$ .

(C) Two-Reflection Model (Szegedy-type)

Given a Markov chain $M$ on $n$ sites, construct projectors $Q_1, Q_2$ onto certain $n$ -dimensional subspaces of $\mathbb{C}^{n^2}$ .
Reflection operators $R_1=2Q_1Q_1^T-I_{n^2}$ , $R_2=2Q_2Q_2^T-I_{n^2}$ .
Evolution: $U_{2r} = R_2 R_1$ (Godsil et al., 2017, Konno et al., 2017).

The models are related via combinatorial correspondences: the arc-reversal model is parameterized by rotation systems (cyclic orderings of local neighbors, corresponding to embeddings of the graph on surfaces), while the shunt-decomposition model corresponds to 1-factorizations of the bipartite cover $K_2 \times X$ .

2. Structural and Spectral Quantities

The limiting behavior of DTQW is characterized by the average mixing matrix: $\overline{M} = \lim_{T\to\infty} \frac{1}{T} \sum_{t=0}^{T-1} (U^t \circ (U^t)^*) = \sum_r F_r \circ F_r,$ where $U = \sum_r e^{i\theta_r} F_r$ is the spectral decomposition and $\circ$ is the Schur (entrywise) product.

Two key descriptors:

Trace: $\operatorname{Tr}(\overline{M}) = \sum_r \operatorname{Tr}(F_r \circ F_r)$ , a measure of return probability (the "stay-home" probability in the long-time limit).
Total entropy: For each column $j$ , define the limiting probability distribution $p^{(j)}$ , then Shannon entropy $H(p^{(j)})=-\sum_i p^{(j)}_i \log p^{(j)}_i$ ; total entropy $S_{\text{tot}}(\overline{M}) = \sum_{j} H(p^{(j)})$ quantifies mixing to uniformity (Godsil et al., 2017).

3. Influence of Graph Structures: Rotation Systems and Factorizations

The arc-reversal DTQW is sensitive to the rotation system, i.e., the embedding of the graph. Distinct cyclic orders at each vertex change the interference patterns. Embeddings of higher genus tend to break global coherence, leading to decreased trace (lower stay-home probability) and higher entropy (better mixing):

Low-genus (planar or toroidal) embeddings favor localization.
High-genus embeddings favor uniform spreading (Godsil et al., 2017).

In the shunt-decomposition model, the walk is governed by the choice of 1-factorization. Symmetric 1-factorizations (each $P_j$ an involution) maximize the trace and minimize mixing; highly asymmetric factorizations enhance mixing at the expense of return probability.

Quantitative exemplars (arc-reversal traces, coin = circulant $3\times 3$ ):

Graph	Genus	$\operatorname{Tr}(\overline{M})$
$K_4$	0	3.000
	1	1.694–1.754
$K_{3,3}$	1	2.111–2.201
	2	1.053
$Q_3$	0	4.500
	2	1.525–1.981

Similar patterns hold for entropy and for the shunt-decomposition case (Godsil et al., 2017).

4. Generalizations, Variants, and Physical Realizations

The algebraic framework for DTQW admits significant variants:

Memory-based or coinless walks: Broader models encode the walk on $\ell^2(X)\otimes\ell^2(X)$ ; the "interchange" or "swap" operator, followed by a set of local unitaries, captures frameworks that generalize coins and support arbitrary local symmetries (Dimcovic et al., 2011).
Projection and dimension reduction: Any DTQW whose step operator respects a partition of vertex space, i.e., with consistency under shift, can be projected to an effective walk on the quotient graph. The resulting walk, possibly with a higher-dimensional coin, inherits spectral and dynamical properties of the parent walk. This principle is formalized in the "Projection Theorem," which also underpins certain optical time-multiplexing implementations (Potoček, 2020).
Graph-directed walks / Multigraphs: Every bipartite quantum circuit can be interpreted as a DTQW on some directed multigraph, and vice versa, due to the algebraic mapping between circuits and block-structured shift operators (Wing-Bocanegra et al., 2023).
Topological and two-dimensional walks: DTQWs on 2D lattices or arbitrary graphs may strongly localize (trapping), or realize nontrivial band structures and topological invariants, depending on coin and shift choices (Kollár et al., 2015).

5. Algorithmic and Physical Significance

DTQWs serve as universal models of quantum computation: any quantum circuit can be efficiently encoded as the evolution of an appropriate (coined) DTQW on a suitable multigraph, with explicit mappings between circuit gates and shift/coin structures. Variants including spatial search (Grover-type) algorithms yield quantum speedups; analysis of return probabilities, mixing times, and localization underpins the design of search and sampling protocols (Wing-Bocanegra et al., 2023, Hamilton et al., 2016).

The sensitivity of DTQWs to combinatorial and topological graph properties has direct impact in optimization, transport phenomena, and simulation of physical equations (e.g., Dirac, Schrödinger) in discretized spacetime (Nzongani et al., 15 Apr 2024, Molfetta et al., 2014). The embedding or factorization structure can be algorithmically tuned to trade between rapid mixing (useful for uniform sampling) and strong localization (useful for targeted searches) (Godsil et al., 2017).

6. Design Strategies and Optimization

Empirical and theoretical analyses reveal that:

Mixing versus localization: To maximize long-term return probability, use low-genus (planar/toroidal) embeddings or symmetric 1-factorizations. To maximize entropy and rapid uniform mixing, employ high-genus rotation systems or highly asymmetric shunt-decompositions (Godsil et al., 2017).
Algorithmic applications: The choice of graph structure and coin acts as a tunable parameter set for balancing coherence, interference, and mixing—providing a flexible substrate for quantum algorithm design.

7. Broader Frameworks and Equivalences

Most DTQW models—including coined, Szegedy (two-reflection), and 2-tessellable staggered models—are unitarily equivalent, intertransformable via two-partition frameworks. They all consist structurally of alternating local unitaries (or reflections) on overlapping partitions of the computational basis, and many spectral properties transfer directly under this unitary equivalence (Konno et al., 2017).

This unification establishes that specific choices of model are a matter of convenience rather than expressive power; the structure is that of a product of two local reflections on suitable combinatorial partitions of the basis set.

References:

(Godsil et al., 2017, Konno et al., 2017, Kollár et al., 2015, Wing-Bocanegra et al., 2023, Dimcovic et al., 2011, Potoček, 2020)